Local existence-uniqueness and continuation of solutions for delay stochastic evolution equations

References

• Caraballo , T . 2001 . Nonlinear partial functional differential equations: Existence and stability . J. Math. Anal. Appl. , 262 : 87 – 111 .
   Web of Science ®Google Scholar
• Liu , Y and Li , Z . 2008 . Stability analysis of zero solution for two classes evolution equations . Nonlin. Anal. , 69 : 2058 – 2076 .
   Web of Science ®Google Scholar
• Lee , H and Alkahby , H . 2008 . Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay . Nonlin. Anal. , 69 : 2158 – 2166 .
   Web of Science ®Google Scholar
• Xie , B . 2008 . On pathwise uniqueness of stochastic evolution equations in Hilbert spaces . J. Math. Anal. Appl. , 344 : 204 – 216 .
   Web of Science ®Google Scholar
• Muthukumar , P and Balasubramaniam , P . 2009 . Approximate controllability of nonlinear stochastic evolution systems with time-varying delays . J. Frank. Institute , 346 : 65 – 80 .
   Web of Science ®Google Scholar
• Luo , J and Liu , K . 2008 . Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps . Stoch. Proc. Appl. , 118 : 396 – 424 .
   Google Scholar
• Taniguchi , T , Liu , K and Truman , A . 2002 . Existence uniqueness and asymptotic behaviour of mild solutions to stochastic functional differential equations in Hilbert spaces . J. Differ. Equ. , 81 : 72 – 91 .
   Google Scholar
• Pardoux , E . 1975 . Doctoral Thesis , Centre d'Orsay : Université de Paris Sud .
   Google Scholar
• Pardoux , E . 1979 . Stochastic partial differential equations and filtering of diffusion processes . Stochastics , 3 : 127 – 167 .
   Google Scholar
• Real , J . 1982 . Stochastic partial differential equations with delays . Stochastics , 8 : 81 – 102 .
   Google Scholar
• Caraballo , T , Liu , K and Truman , A . 2000 . Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property . Proc. R. Soc. Lond. A , 456 : 1775 – 1802 .
   Web of Science ®Google Scholar
• Caraballo , T , Carrido-Atienza , MJ and Real , J . 2002 . Existence and uniqueness of solutions for delay stochastic evolution equations . Stoch. Anal. Appl. , 20 : 1225 – 1256 .
   Web of Science ®Google Scholar
• Miller , RK and Michel , AN . 1982 . Ordinary Differential Equations , New York : Academic Press .
   Google Scholar
• Philp , H . 1982 . Ordinary Differential Equations, . , 2nd ed , New York : Birkhäuser .
   Google Scholar
• Friedman , A . 1975 . Stochastic Differential Equations and Applications , New York : Academic Press .
   Google Scholar
• Mao , X . 1997 . Stochastic Differential Equations and Applications , Chichester, , UK : Horwood Publishing .
   Google Scholar
• Xu , D , Yang , Z and Huang , Y . 2008 . Existence-uniqueness and continuation theorems for stochastic functional differential equations . J. Diff. Equ. , 245 : 1681 – 1703 .
   Web of Science ®Google Scholar
• Driver , RD . 1977 . Ordinary and Delay Differential Equations , New York : Springer-Verlag .
   Google Scholar
• Hale , JK and Lunel , SMV . 1993 . Introduction to Functional Differential Equations , New York : Springer-Verlag .
   Google Scholar
• Taniguchi , T . 1992 . Successive approximations to solutions of stochastic differential equations . J. Differ. Equ. , 96 : 152 – 169 .
   Web of Science ®Google Scholar
• Shen , Y , Luo , Q and Mao , X . 2006 . The improved LaSalle-type theorems for stochastic functional differential equations . J. Math. Anal. Appl. , 318 : 134 – 154 .
   Web of Science ®Google Scholar
• Métivier , M and Pellaumail , J . 1980 . Stochastic Integration , New York : Academic Press .
   Google Scholar
• Taniguchi , T . 1990 . On sufficient conditions for non-explosion of solutions to stochastic differential equations . J. Math. Anal. Appl. , 153 : 549 – 561 .
   Google Scholar